Nonlinear forced vibration of functionally graded carbon nanotube reinforced composite circular cylindrical shells

Abstract

The nonlinear forced vibration characteristics of functionally graded carbon nanotube reinforced composite (FG-CNTRC) circular cylindrical shells are investigated. On the basis of Reddy’s first-order shear deformation theory, von Kármán geometric nonlinearity and Hamilton’s principle, the equations of motion are derived. The Galerkin technique is applied to discretize the partial differential equations into nonlinear ordinary differential equations, which are reduced by using Volmir’s assumption and the static condensation method. The incremental harmonic balance method is applied to analyze the dynamic response of FG-CNTRC cylindrical shells. A convergence study on the mode expansions is conducted by considering both axisymmetric and asymmetric modes. The natural frequencies and the resonance responses are compared with existing studies to examine the validity of this study. The effects of distribution and volume fraction of carbon nanotube, thickness-to-radius ratio, length-to-radius ratio, dimensionless radial excitation amplitude and dam** ratio on the resonance responses of FG-CNTRC cylindrical shells are discussed. The results show that the reduced model of the system is reasonable. The frequency responses of FG-CNTRC cylindrical shells show both hardening and softening types of nonlinearities, and they are greatly influenced by the change of the fundamental vibrational mode.

Appendix A

Rewrite Eq. (23) in vector form as

$$\begin{aligned} \begin{aligned}&u(x,\theta ,t)={\mathbf {U}}^{{\mathrm {T}}}(x,\theta ){\mathbf {q}}_{u} (t), \quad v(x,\theta ,t)={\mathbf {V}}^{{\mathrm {T}}}(x,\theta ){\mathbf {q}}_{v} (t), \quad w(x,\theta ,t)={\mathbf {W}}^{{\mathrm {T}}}(x,\theta ){\mathbf {q}}_{w} (t),\\ {}&\phi _{x} (x,\theta ,t)={{{\varvec{\Phi }} }}_{x}^{\mathrm {T}} (x,\theta ){\mathbf {q}}_{\phi _{x} } (t), \quad \phi _{\theta } (x,\theta ,t)={{{\varvec{\Phi }} }}_{\theta }^{\mathrm {T}} (x,\theta ){\mathbf {q}}_{\phi _{\theta } } (t), \end{aligned} \end{aligned}$$

where $U_{m,jn} (x,\theta )=\varPhi _{xm,jn} (x,\theta )=\cos (m\pi x)\cos (jn\theta )$, $V_{m,jn} (x,\theta )=\varPhi _{\theta m,jn} (x,\theta )=\sin (m\pi x)\sin (jn\theta )$ and $W_{m,jn} (x,\theta )=\sin (m\pi x)\cos (jn\theta )$.

The elements of the matrix ${\mathbf{M}}$ are

$$\begin{aligned} \begin{aligned} {\mathbf {M}}_{\mathrm {11}}=&{} \int _0^L {\int _0^{2\pi } {I_{0} {\mathbf {UU}}^{\mathrm {T}}} } \text{ d } \theta \, \text{ d } x, \quad {\mathbf {M}}_{\mathrm {14}} =\int _0^L {\int _0^{2\pi } {I_{1} {\mathbf {U}}{\varvec{\Phi }} _{x}^{\mathrm {T}} } } \text{ d } \theta \,\text{ d } x;\\ {\mathbf {M}}_{\mathrm {22}}=&{} \int _0^L {\int _0^{2\pi } {I_{0} {\mathbf {VV}}^{\mathrm {T}}} } \text{ d } \theta \,\text{ d } x, \quad {\mathbf {M}}_{\mathrm {25}} =\int _0^L \int _0^{2\pi } {I_{1} \mathbf {V}{\varvec{\Phi }}_{\theta }^{\mathrm {T}}} \text{ d } \theta \,\text{ d }\,x,\\ {\mathbf {M}}_{\mathrm {33}}=&{} \int _0^L {\int _0^{2\pi } {I_{0} {\mathbf {WW}}^{\mathrm {T}}} } \text{ d } \theta \,\text{ d }\,x;\\ {\mathbf {M}}_{\mathrm {41}}=&{} \int _0^L {\int _0^{2\pi } {I_{1} {{{\varvec{\Phi }}}}_{x} {\mathbf {U}}^{{\mathrm {T}}}} } \text{ d } \theta \,\text{ d }\,x, \quad {\mathbf {M}}_{\mathrm {44}} =\int _0^L {\int _0^{2\pi } {I_{2} {{{\varvec{\Phi }} }}_{x} {{{\varvec{\Phi }} }}_{x}^{\mathrm {T}} } } \text{ d }\,\theta \,\text{ d }\,x;\\ {\mathbf {M}}_{\mathrm {52}}=&{} \int _0^L {\int _0^{2\pi } {I_{1} {{{\varvec{\Phi }} }}_{\theta } {\mathbf {V}}^{\mathrm {T}}} } \text{ d }\,\theta \, \text{ d }\,x, \quad {\mathbf {M}}_{\mathrm {55}} =\int _0^L {\int _0^{2\pi } {I_{2} {{{\varvec{\Phi }} }}_{\theta } {{{\varvec{\Phi }} }}_{\theta }^{\mathrm {T}} } } \text{ d }\,\theta \,\text{ d }\,x. \end{aligned} \end{aligned}$$

The elements of the matrix ${\mathbf{K}}$ are

$$\begin{aligned} {\mathbf{K}}_{\mathrm{11}}= & {} -\int _0^1 {\int _0^{2\pi } {\left( {a_{11} {\mathbf{U}}\frac{\partial ^{2}{\mathbf{U}}^{\mathrm{T}}}{\partial x^{2}}+a_{66} \lambda _{1}^{2} {\mathbf{U}}\frac{\partial ^{2}{\mathbf{U}}^{\mathrm{T}}}{\partial \theta ^{2}}} \right) } } \hbox {d}\,\theta \,\hbox {d}\,x, \quad {\mathbf{K}}_{\mathrm{12}} =-\left( {a_{12} +a_{66} } \right) \lambda _{1} \int _0^1 {\int _0^{2\pi } {{\mathbf{U}}\frac{\partial ^{2}{\mathbf{V}}^{\mathrm{T}}}{\partial x\partial \theta }} } \hbox {d}\,\theta \,\hbox {d}\,x,\\ {\mathbf{K}}_{\mathrm{13}}= & {} -a_{12} \lambda _{1} \int _0^1 {\int _0^{2\pi } {{\mathbf{U}}\frac{\partial {\mathbf{W}}^{\mathrm{T}}}{\partial x}} } \hbox {d}\,\theta \,\hbox {d}\,x, \quad {\mathbf{K}}_{\mathrm{14}} =-\int _0^1 {\int _0^{2\pi } {\left( {b_{11} {\mathbf{U}}\frac{\partial ^{2}{{{\varvec{\Phi }} }}_{x}^{\mathrm{T}} }{\partial x^{2}}+b_{66} \lambda _{1}^{2} {\mathbf{U}}\frac{\partial ^{2}{{{\varvec{\Phi }} }}_{x}^{\mathrm{T}} }{\partial \theta ^{2}}} \right) } } \hbox {d}\,\theta \,\hbox {d}\,x,\\ {\mathbf{K}}_{\mathrm{15}}= & {} -\left( {b_{12} +b_{66} } \right) \lambda _{1} \int _0^1 {\int _0^{2\pi } {{\mathbf{U}}\frac{\partial ^{2}{{{\varvec{\Phi }} }}_{\theta }^{\mathrm{T}} }{\partial x\partial \theta }} } \hbox {d}\,\theta \,\hbox {d}\,x;\\ {\mathbf{K}}_{\mathrm{21}}= & {} -\left( {a_{12} +a_{66} } \right) \lambda _{1} \int _0^1 {\int _0^{2\pi } {{\mathbf{V}}\frac{\partial ^{2}{\mathbf{U}}^{\mathrm{T}}}{\partial x\partial \theta }} } \hbox {d}\,\theta \,\hbox {d}\,x, \quad {\mathbf{K}}_{\mathrm{22}} =-\int _0^1 {\int _0^{2\pi } {\left( {a_{66} {\mathbf{V}}\frac{\partial ^{2}{\mathbf{V}}^{\mathrm{T}}}{\partial x^{2}}+a_{22} \lambda _{1}^{2} {\mathbf{V}}\frac{\partial ^{2}{\mathbf{V}}^{\mathrm{T}}}{\partial \theta ^{2}}} \right) } } \hbox {d}\,\theta \,\hbox {d}\,x,\\ {\mathbf{K}}_{\mathrm{23}}= & {} -a_{22} \lambda _{1}^{2} \int _0^1 {\int _0^{2\pi } {{\mathbf{V}}\frac{\partial {\mathbf{W}}^{\mathrm{T}}}{\partial \theta }} } \hbox {d}\,\theta \,\hbox {d}\,x, \quad {\mathbf{K}}_{\mathrm{24}} =-\left( {b_{12} +b_{66} } \right) \lambda _{1} \int _0^1 {\int _0^{2\pi } {{\mathbf{V}}\frac{\partial ^{2}{{{\varvec{\Phi }} }}_{x}^{\mathrm{T}} }{\partial x\partial \theta }} } \hbox {d}\,\theta \,\hbox {d}\,x,\\ {\mathbf{K}}_{\mathrm{25}}= & {} -\int _0^1 {\int _0^{2\pi } {\left( {b_{66} {\mathbf{V}}\frac{\partial ^{2}{{{\varvec{\Phi }} }}_{\theta }^{\mathrm{T}} }{\partial x^{2}}+b_{22} \lambda _{1}^{2} {\mathbf{V}}\frac{\partial ^{2}{{{\varvec{\Phi }} }}_{\theta }^{\mathrm{T}} }{\partial \theta ^{2}}} \right) } } \hbox {d}\,\theta \,\hbox {d}\,x;\\ {\mathbf{K}}_{\mathrm{31}}= & {} a_{12} \lambda _{1} \int _0^1 {\int _0^{2\pi } {{\mathbf{W}}\frac{\partial {\mathbf{U}}^{\mathrm{T}}}{\partial x}} } \hbox {d}\,\theta \,\hbox {d}\,x, \quad {\mathbf{K}}_{\mathrm{32}} =a_{22} \lambda _{1}^{2} \int _0^1 {\int _0^{2\pi } {{\mathbf{W}}\frac{\partial {\mathbf{V}}^{\mathrm{T}}}{\partial \theta }} } \hbox {d}\,\theta \,\hbox {d}\,x,\\ {\mathbf{K}}_{\mathrm{33}}= & {} -\int _0^1 {\int _0^{2\pi } {\left( {a_{55} {\mathbf{W}}\frac{\partial ^{2}{\mathbf{W}}^{\mathrm{T}}}{\partial x^{2}}+a_{44} \lambda _{1}^{2} {\mathbf{W}}\frac{\partial ^{2}{\mathbf{W}}^{\mathrm{T}}}{\partial \theta ^{2}}-a_{22} \lambda _{1}^{2} {\mathbf{WW}}^{\mathrm{T}}} \right) } } \hbox {d}\,\theta \,\hbox {d}\,x,\\ {\mathbf{K}}_{\mathrm{34}}= & {} -\left( {\frac{a_{55} }{\lambda _{3} }-b_{12} \lambda _{1} } \right) \int _0^1 {\int _0^{2\pi } {{\mathbf{W}}\frac{\partial {{{\varvec{\Phi }} }}_{x}^{\mathrm{T}} }{\partial x}} } \hbox {d}\,\theta \,\hbox {d}\,x,\\ {\mathbf{K}}_{\mathrm{35}}= & {} -\left( {a_{44} \frac{\lambda _{1} }{\lambda _{3} }-b_{22} \lambda _{1}^{2} } \right) \int _0^1 \int _0^{2\pi } {{\mathbf{W}}{\varvec{\Phi }}_{\theta } \frac{\partial {{{\varvec{\Phi }} }}_{\theta }^{\mathrm{T}} }{\partial \theta }} \hbox {d}\,\theta \,\hbox {d}\,x;\\ {\mathbf{K}}_{\mathrm{41}}= & {} -\int _0^1 {\int _0^{2\pi } {\left( {b_{11} {{{\varvec{\Phi }} }}_{x} \frac{\partial ^{2}{\mathbf{U}}^{\mathrm{T}}}{\partial x^{2}}+b_{66} \lambda _{1}^{2} {{{\varvec{\Phi }} }}_{x} \frac{\partial ^{2}{\mathbf{U}}^{\mathrm{T}}}{\partial \theta ^{2}}} \right) } } \hbox {d}\,\theta \,\hbox {d}\,x, \\ {\mathbf{K}}_{\mathrm{42}}= & {} -\left( {b_{12} +b_{66} } \right) \lambda _{1} \int _0^1 {\int _0^{2\pi } {{{{\varvec{\Phi }} }}_{x} \frac{\partial ^{2}{\mathbf{V}}^{\mathrm{T}}}{\partial x\partial \theta }} } \hbox {d}\,\theta \,\hbox {d}\,x, \\ {\mathbf{K}}_{\mathrm{43}}= & {} -\left( {b_{12} \lambda _{1} -\frac{a_{55} }{\lambda _{3} }} \right) \int _0^1 {\int _0^{2\pi } {{{{\varvec{\Phi }} }}_{x} \frac{\partial {\mathbf{W}}^{\mathrm{T}}}{\partial x}} } \hbox {d}\,\theta \,\hbox {d}\,x,\\ {\mathbf{K}}_{\mathrm{44}}= & {} -\int _0^1 {\int _0^{2\pi } {\left( {d_{11} {{{\varvec{\Phi }} }}_{x} \frac{\partial ^{2}{{{\varvec{\Phi }} }}_{x}^{\mathrm{T}} }{\partial x^{2}}+d_{66} \lambda _{1}^{2} {{{\varvec{\Phi }} }}_{x} \frac{\partial ^{2}{{{\varvec{\Phi }} }}_{x}^{\mathrm{T}} }{\partial \theta ^{2}}-\frac{a_{55} }{\lambda _{3}^{2} }{{{\varvec{\Phi }} }}_{x} {{{\varvec{\Phi }} }}_{x}^{\mathrm{T}} } \right) } } \hbox {d}\,\theta \,\hbox {d}\,x,\\ {\mathbf{K}}_{\mathrm{45}}= & {} -\left( {d_{12} +d_{66} } \right) \lambda _{1} \int _0^1 {\int _0^{2\pi } {{{{\varvec{\Phi }} }}_{x} \frac{\partial ^{2}{{{\varvec{\Phi }} }}_{\theta }^{\mathrm{T}} }{\partial x\partial \theta }} } \hbox {d}\,\theta \,\hbox {d}\,x;\\ {\mathbf{K}}_{\mathrm{51}}= & {} -\left( {b_{12} +b_{66} } \right) \lambda _{1} \int _0^1 {\int _0^{2\pi } {{{{\varvec{\Phi }} }}_{\theta } \frac{\partial ^{2}{\mathbf{U}}^{\mathrm{T}}}{\partial x\partial \theta }} } \hbox {d}\,\theta \,\hbox {d}\,x, \\ {\mathbf{K}}_{\mathrm{52}}= & {} -\int _0^1 {\int _0^{2\pi } {\left( {b_{66} {{{\varvec{\Phi }} }}_{\theta } \frac{\partial ^{2}{\mathbf{V}}^{\mathrm{T}}}{\partial x^{2}}+b_{22} \lambda _{1}^{2} {{{\varvec{\Phi }}}}_{\theta } \frac{\partial ^{2}{\mathbf{V}}^{\mathrm{T}}}{\partial \theta ^{2}}} \right) } } \hbox {d}\,\theta \,\hbox {d}\,x, \\ {\mathbf{K}}_{\mathrm{53}}= & {} -\left( {b_{22} \lambda _{1}^{2} -a_{44} \frac{\lambda _{1} }{\lambda _{3} }} \right) \int _0^1 {\int _0^{2\pi } {{{{\varvec{\Phi }} }}_{\theta } \frac{\partial {\mathbf{W}}^{\mathrm{T}}}{\partial \theta }} } \hbox {d}\,\theta \,\hbox {d}\,x, \\ {\mathbf{K}}_{\mathrm{54}}= & {} -\left( {d_{12} +d_{66} } \right) \lambda _{1} \int _0^1 {\int _0^{2\pi } {{{{\varvec{\Phi }} }}_{\theta } \frac{\partial ^{2}{{{\varvec{\Phi }} }}_{x}^{\mathrm{T}} }{\partial x\partial \theta }} } \hbox {d}\,\theta \,\hbox {d}\,x, \\ {\mathbf{K}}_{\mathrm{55}}= & {} -\int _0^1 {\int _0^{2\pi } {\left( {d_{66} {{{\varvec{\Phi }} }}_{\theta } \frac{\partial ^{2}{{{\varvec{\Phi }} }}_{\theta }^{\mathrm{T}} }{\partial x^{2}}+d_{22} \lambda _{1}^{2} {{{\varvec{\Phi }} }}_{\theta } \frac{\partial ^{2}{{{\varvec{\Phi }} }}_{\theta }^{\mathrm{T}} }{\partial \theta ^{2}}-\frac{a_{44} }{\lambda _{3}^{2} }{{{\varvec{\Phi }} }}_{\theta } {{{\varvec{\Phi }} }}_{\theta }^{\mathrm{T}} } \right) } } \hbox {d}\,\theta \,\hbox {d}\,x. \end{aligned}$$

The elements of the matrix ${\mathbf{K}}^{(2)}$ are

$$\begin{aligned} {\mathbf{K}}_{13}^{(2)}&=-\int _0^1 {\int _0^{2\pi } {{\mathbf{U}}\frac{\partial {\mathbf{W}}^{\mathrm{T}}}{\partial x}{\mathbf{q}}_{w} \left( {a_{11} \lambda _{3} \frac{\partial ^{2}{\mathbf{W}}^{\mathrm{T}}}{\partial x^{2}}+a_{66} \lambda _{1} \lambda _{2} \frac{\partial ^{2}{\mathbf{W}}^{\mathrm{T}}}{\partial \theta ^{2}}} \right) } } \hbox {d}\,\theta \,\hbox {d}\,x \\&\quad -\left( {a_{12} +a_{66} } \right) \lambda _{1} \lambda _{2} \int _0^1 {\int _0^{2\pi } {{\mathbf{U}}\frac{\partial {\mathbf{W}}^{\mathrm{T}}}{\partial \theta }{\mathbf{q}}_{w} \frac{\partial ^{2}{\mathbf{W}}^{\mathrm{T}}}{\partial x\partial \theta }} } \hbox {d}\,\theta \,\hbox {d}\,x ;\\ {\mathbf{K}}_{23}^{(2)}&=-\left( {a_{12} +a_{66} } \right) \lambda _{2} \int _0^1 {\int _0^{2\pi } {{\mathbf{V}}\frac{\partial {\mathbf{W}}^{\mathrm{T}}}{\partial x}{\mathbf{q}}_{w} \frac{\partial ^{2}{\mathbf{W}}^{\mathrm{T}}}{\partial x\partial \theta }} } \hbox {d}\,\theta \,\hbox {d}\,x \\&\quad -\int _0^1 {\int _0^{2\pi } {{\mathbf{V}}\frac{\partial {\mathbf{W}}^{\mathrm{T}}}{\partial \theta }{\mathbf{q}}_{w} \left( {a_{66} \lambda _{2} \frac{\partial ^{2}{\mathbf{W}}^{\mathrm{T}}}{\partial x^{2}}+a_{22} \lambda _{1}^{2} \lambda _{2} \frac{\partial ^{2}{\mathbf{W}}^{\mathrm{T}}}{\partial \theta ^{2}}} \right) } } \hbox {d}\,\theta \,\hbox {d}\,x ;\\ {\mathbf{K}}_{31}^{(2)}&=-\int _0^1 {\int _0^{2\pi } {{\mathbf{W}}\frac{\partial {\mathbf{W}}^{\mathrm{T}}}{\partial x}{\mathbf{q}}_{w} \left( {a_{11} \lambda _{3} \frac{\partial ^{2}{\mathbf{U}}^{\mathrm{T}}}{\partial x^{2}}+a_{66} \lambda _{1} \lambda _{2} \frac{\partial ^{2}{\mathbf{U}}^{\mathrm{T}}}{\partial \theta ^{2}}} \right) } } \hbox {d}\,\theta \,\hbox {d}\,x \\&\quad -\left( {a_{12} +a_{66} } \right) \lambda _{1} \lambda _{2} \int _0^1 {\int _0^{2\pi } {{\mathbf{W}}\frac{\partial {\mathbf{W}}^{\mathrm{T}}}{\partial \theta }{\mathbf{q}}_{w} \frac{\partial ^{2}{\mathbf{U}}^{\mathrm{T}}}{\partial x\partial \theta }} } \hbox {d}\,\theta \,\hbox {d}\,x \\&\quad -\int _0^1 {\int _0^{2\pi } {{\mathbf{W}}\left( {a_{11} \lambda _{3} \frac{\partial ^{2}{\mathbf{W}}^{\mathrm{T}}}{\partial x^{2}}+a_{12} \lambda _{1} \lambda _{2} \frac{\partial ^{2}{\mathbf{W}}^{\mathrm{T}}}{\partial \theta ^{2}}} \right) } } {\mathbf{q}}_{w} \frac{\partial {\mathbf{U}}^{\mathrm{T}}}{\partial x}\hbox {d}\,\theta \,\hbox {d}\,x \\&\quad -2a_{66} \lambda _{1} \lambda _{2} \int _0^1 {\int _0^{2\pi } {{\mathbf{W}}\frac{\partial ^{2}{\mathbf{W}}^{\mathrm{T}}}{\partial x\partial \theta }} } {\mathbf{q}}_{w} \frac{\partial {\mathbf{U}}^{\mathrm{T}}}{\partial \theta }\hbox {d}\,\theta \,\hbox {d}\,x ,\\ {\mathbf{K}}_{32}^{(2)}&=-\left( {a_{12} +a_{66} } \right) \lambda _{2} \int _0^1 {\int _0^{2\pi } {{\mathbf{W}}\frac{\partial {\mathbf{W}}^{\mathrm{T}}}{\partial x}{\mathbf{q}}_{w} \frac{\partial ^{2}{\mathbf{V}}^{\mathrm{T}}}{\partial x\partial \theta }} } \hbox {d}\,\theta \,\hbox {d}\,x \\&\quad -\int _0^1 {\int _0^{2\pi } {{\mathbf{W}}\frac{\partial {\mathbf{W}}^{\mathrm{T}}}{\partial \theta }{\mathbf{q}}_{w} \left( {a_{66} \lambda _{2} \frac{\partial ^{2}{\mathbf{V}}^{\mathrm{T}}}{\partial x^{2}}+a_{22} \lambda _{1}^{2} \lambda _{2} \frac{\partial ^{2}{\mathbf{V}}^{\mathrm{T}}}{\partial \theta ^{2}}} \right) } } \hbox {d}\,\theta \,\hbox {d}\,x \\&\quad -2a_{66} \lambda _{2} \int _0^1 {\int _0^{2\pi } {{\mathbf{W}}\frac{\partial ^{2}{\mathbf{W}}^{\mathrm{T}}}{\partial x\partial \theta }} } {\mathbf{q}}_{w} \frac{\partial {\mathbf{V}}^{\mathrm{T}}}{\partial x}\hbox {d}\,\theta \,\hbox {d}\,x\\&\quad -\int _0^1 {\int _0^{2\pi } {{\mathbf{W}}\left( {a_{12} \lambda _{2} \frac{\partial ^{2}{\mathbf{W}}^{\mathrm{T}}}{\partial x^{2}}+a_{22} \lambda _{1}^{2} \lambda _{2} \frac{\partial ^{2}{\mathbf{W}}^{\mathrm{T}}}{\partial \theta ^{2}}} \right) } } {\mathbf{q}}_{w} \frac{\partial {\mathbf{V}}^{\mathrm{T}}}{\partial \theta }\hbox {d}\,\theta \,\hbox {d}\,x,\\ {\mathbf{K}}_{33}^{(2)}&=-\frac{a_{12} }{2}\lambda _{2} \int _0^1 {\int _0^{2\pi } {{\mathbf{W}}\frac{\partial {\mathbf{W}}^{\mathrm{T}}}{\partial x}{\mathbf{q}}_{w} \frac{\partial {\mathbf{W}}^{\mathrm{T}}}{\partial x}} } \hbox {d}\,\theta \,\hbox {d}\,x-\frac{a_{22} }{2}\lambda _{1}^{2} \lambda _{2} \int _0^1 {\int _0^{2\pi } {{\mathbf{W}}\frac{\partial {\mathbf{W}}^{\mathrm{T}}}{\partial \theta }{\mathbf{q}}_{w} \frac{\partial {\mathbf{W}}^{\mathrm{T}}}{\partial \theta }} } \hbox {d}\,\theta \,\hbox {d}\,x \\&\quad -\int _0^1 {\int _0^{2\pi } {{\mathbf{W}}\left( {a_{12} \lambda _{2} \frac{\partial ^{2}{\mathbf{W}}^{\mathrm{T}}}{\partial x^{2}}+a_{22} \lambda _{1}^{2} \lambda _{2} \frac{\partial ^{2}{\mathbf{W}}^{\mathrm{T}}}{\partial \theta ^{2}}} \right) } } {\mathbf{q}}_{w} {\mathbf{W}}^{\mathrm{T}}\hbox {d}\,\theta \,\hbox {d}\,x ,\\ {\mathbf{K}}_{34}^{(2)}&=-\int _0^1 {\int _0^{2\pi } {{\mathbf{W}}\frac{\partial {\mathbf{W}}^{\mathrm{T}}}{\partial x}{\mathbf{q}}_{w} \left( {b_{11} \lambda _{3} \frac{\partial ^{2}{{{\varvec{\Phi }} }}_{x}^{\mathrm{T}} }{\partial x^{2}}+b_{66} \lambda _{1} \lambda _{2} \frac{\partial ^{2}{{{\varvec{\Phi }} }}_{x}^{\mathrm{T}} }{\partial \theta ^{2}}} \right) } } \hbox {d}\,\theta \,\hbox {d}\,x \\&\quad -\left( {b_{12} +b_{66} } \right) \lambda _{1} \lambda _{2} \int _0^1 {\int _0^{2\pi } {{\mathbf{W}}\frac{\partial {\mathbf{W}}^{\mathrm{T}}}{\partial \theta }{\mathbf{q}}_{w} \frac{\partial ^{2}{{{\varvec{\Phi }} }}_{x}^{\mathrm{T}} }{\partial x\partial \theta }} } \hbox {d}\,\theta \,\hbox {d}\,x \\&\quad -\int _0^1 {\int _0^{2\pi } {{\mathbf{W}}\left( {b_{11} \lambda _{3} \frac{\partial ^{2}{\mathbf{W}}^{\mathrm{T}}}{\partial x^{2}}+b_{12} \lambda _{1} \lambda _{2} \frac{\partial ^{2}{\mathbf{W}}^{\mathrm{T}}}{\partial \theta ^{2}}} \right) } } {\mathbf{q}}_{w} \frac{\partial {{{\varvec{\Phi }} }}_{x}^{\mathrm{T}} }{\partial x}\hbox {d}\,\theta \,\hbox {d}\,x\\&\quad -2b_{66} \lambda _{1} \lambda _{2} \int _0^1 {\int _0^{2\pi } {{\mathbf{W}}\frac{\partial ^{2}{\mathbf{W}}^{\mathrm{T}}}{\partial x\partial \theta }{\mathbf{q}}_{w} } } \frac{\partial {{{\varvec{\Phi }} }}_{x}^{\mathrm{T}} }{\partial \theta }\hbox {d}\,\theta \,\hbox {d}\,x ,\\ {\mathbf{K}}_{35}^{(2)}&=-\left( {b_{12} +b_{66} } \right) \lambda _{2} \int _0^1 {\int _0^{2\pi } {{\mathbf{W}}\frac{\partial {\mathbf{W}}^{\mathrm{T}}}{\partial x}{\mathbf{q}}_{w} \frac{\partial ^{2}{{{\varvec{\Phi }} }}_{\theta }^{\mathrm{T}} }{\partial x\partial \theta }} } \hbox {d}\,\theta \,\hbox {d}\,x \\&\quad -\int _0^1 {\int _0^{2\pi } {{\mathbf{W}}\frac{\partial {\mathbf{W}}^{\mathrm{T}}}{\partial \theta }{\mathbf{q}}_{w} \left( {b_{66} \lambda _{2} \frac{\partial ^{2}{{{\varvec{\Phi }} }}_{\theta }^{\mathrm{T}} }{\partial x^{2}}+b_{22} \lambda _{1}^{2} \lambda _{2} \frac{\partial ^{2}{{{\varvec{\Phi }} }}_{\theta }^{\mathrm{T}} }{\partial \theta ^{2}}} \right) } } \hbox {d}\,\theta \,\hbox {d}\,x \\&\quad -2b_{66} \lambda _{2} \int _0^1 {\int _0^{2\pi } {{\mathbf{W}}\frac{\partial ^{2}{\mathbf{W}}^{\mathrm{T}}}{\partial x\partial \theta }{\mathbf{q}}_{w} } } \frac{\partial {{{\varvec{\Phi }} }}_{\theta }^{\mathrm{T}} }{\partial x}\hbox {d}\,\theta \,\hbox {d}\,x\\&\quad -\int _0^1 {\int _0^{2\pi } {{\mathbf{W}}\left( {b_{12} \lambda _{2} \frac{\partial ^{2}{\mathbf{W}}^{\mathrm{T}}}{\partial x^{2}}+b_{22} \lambda _{1}^{2} \lambda _{2} \frac{\partial ^{2}{\mathbf{W}}^{\mathrm{T}}}{\partial \theta ^{2}}} \right) } } {\mathbf{q}}_{w} \frac{\partial {{{\varvec{\Phi }} }}_{\theta }^{\mathrm{T}} }{\partial \theta }\hbox {d}\,\theta \,\hbox {d}\,x ;\\ {\mathbf{K}}_{43}^{(2)}&=-\int _0^1 {\int _0^{2\pi } {{{{\varvec{\Phi }} }}_{x} \frac{\partial {\mathbf{W}}^{\mathrm{T}}}{\partial x}{\mathbf{q}}_{w} \left( {b_{11} \lambda _{3} \frac{\partial ^{2}{\mathbf{W}}^{\mathrm{T}}}{\partial x^{2}}+b_{66} \lambda _{1} \lambda _{2} \frac{\partial ^{2}{\mathbf{W}}^{\mathrm{T}}}{\partial \theta ^{2}}} \right) } } \hbox {d}\,\theta \,\hbox {d}\,x \\&\quad -\left( {b_{12} +b_{66} } \right) \lambda _{1} \lambda _{2} \int _0^1 {\int _0^{2\pi } {{{{\varvec{\Phi }} }}_{x} \frac{\partial {\mathbf{W}}^{\mathrm{T}}}{\partial \theta }{\mathbf{q}}_{w} \frac{\partial ^{2}\mathrm{\mathbf{W}}^{\mathrm{T}}}{\partial x\partial \theta }} } \hbox {d}\,\theta \,\hbox {d}\,x ;\\ {\mathbf{K}}_{53}^{(2)}&=-\left( {b_{12} +b_{66} } \right) \lambda _{2} \int _0^1 {\int _0^{2\pi } {{{{\varvec{\Phi }} }}_{\theta } \frac{\partial {\mathbf{W}}^{\mathrm{T}}}{\partial x}{\mathbf{q}}_{w} \frac{\partial ^{2}{\mathbf{W}}^{\mathrm{T}}}{\partial x\partial \theta }} } \hbox {d}\,\theta \,\hbox {d}\,x \\&\quad -\int _0^1 {\int _0^{2\pi } {{{{\varvec{\Phi }} }}_{\theta } \frac{\partial {\mathbf{W}}^{\mathrm{T}}}{\partial \theta }{\mathbf{q}}_{w} \left( {b_{66} \lambda _{2} \frac{\partial ^{2}{\mathbf{W}}^{\mathrm{T}}}{\partial x^{2}}+b_{22} \lambda _{1}^{2} \lambda _{2} \frac{\partial ^{2}{\mathbf{W}}^{\mathrm{T}}}{\partial \theta ^{2}}} \right) } } \hbox {d}\,\theta \,\hbox {d}\,x . \end{aligned}$$

The elements of the matrix ${\mathbf{K}}^{(3)}$ are

$$\begin{aligned} \begin{aligned} {\mathbf{K}}_{33}^{(3)}&=-\int _0^1 {\int _0^{2\pi } {{\mathbf{W}}\left( {\frac{3a_{11} }{2}\lambda _{3}^{2} \frac{\partial ^{2}{\mathbf{W}}^{\mathrm{T}}}{\partial x^{2}}+\frac{a_{12} +2a_{66} }{2}\lambda _{2}^{2} \frac{\partial ^{2}{\mathbf{W}}^{\mathrm{T}}}{\partial \theta ^{2}}} \right) {\mathbf{q}}_{w} \frac{\partial {\mathbf{W}}^{\mathrm{T}}}{\partial x}{\mathbf{q}}_{w} \frac{\partial {\mathbf{W}}^{\mathrm{T}}}{\partial x}} } \hbox {d}\,\theta \,\hbox {d}\,x \\&\quad -\int _0^1 {\int _0^{2\pi } {{\mathbf{W}}\left( {\frac{a_{12} +2a_{66} }{2}\lambda _{2}^{2} \frac{\partial ^{2}{\mathbf{W}}^{\mathrm{T}}}{\partial x^{2}}+\frac{3a_{22} }{2}\lambda _{1}^{2} \lambda _{2}^{2} \frac{\partial ^{2}{\mathbf{W}}^{\mathrm{T}}}{\partial \theta ^{2}}} \right) } } {\mathbf{q}}_{w} \frac{\partial {\mathbf{W}}^{\mathrm{T}}}{\partial \theta }{\mathbf{q}}_{w} \frac{\partial {\mathbf{W}}^{\mathrm{T}}}{\partial \theta }\hbox {d}\,\theta \,\hbox {d}\,x \\&\quad -\left( {2a_{12} +4a_{66} } \right) \lambda _{2}^{2} \int _0^1 {\int _0^{2\pi } {{\mathbf{W}}\frac{\partial {\mathbf{W}}^{\mathrm{T}}}{\partial x}{\mathbf{q}}_{w} \frac{\partial {\mathbf{W}}^{\mathrm{T}}}{\partial \theta }{\mathbf{q}}_{w} \frac{\partial ^{2}{\mathbf{W}}^{\mathrm{T}}}{\partial x\partial \theta }} } \hbox {d}\,\theta \,\hbox {d}\,x. \end{aligned} \end{aligned}$$